Significant progress has been made not only in improving the yields of carbon nanotubes, but also in gaining a profound fundamental understanding of the growth processes. Flames are emerging as a powerful tool for the synthesis of carbon nanotubes and carbon nanofibers. The flame volume provides a carbon-rich chemically reactive environment capable of generating nanostructures during short residence times in a continuous single-step process. The present work provides a concise review of the advances made over the past two decades in the areas of flame synthesis of carbon nanotubes and carbon nanofibers. An overview of existing flame methods to synthesize carbon nanotubes is first provided. Various catalytic materials, fuel types, and flame configurations have been employed in an attempt to achieve controlled synthesis of carbon nanotubes and carbon nanofibers. Diffusion and premixed flames in counter-flow and co-flow geometries are also discussed. Various hydrocarbon fuels, oxygen enrichment, and dilution with inert gases are then examined in detail. The ability to synthesize and control carbon nanotubes and carbon nanofibers is essential for the fabrication of nanomechanical and electrical devices. A fundamental understanding of the growth mechanism and development of control methods is critically important to address these issues. The purpose of the present review is to clarify the growth mechanisms and achieve controlled flame synthesis of carbon nanotubes and carbon nanofibers.